marlon f. sacedon dept. of mathematics, physics, statistics (dmps)

Marlon F. SacedonDept. of Mathematics, Physics, Statistics (DMPS)

College of Arts and Sciences, Visayas State UniversityVisca, Baybay City, Leyte, Philippines

GENERALIZED MATHEMATICAL FORMULA FOR DYNAMICS ON ONE-

DIMENSIONAL MOTION OF TWO-OBJECT SYSTEM

Formerly ViSCA

Baybay City, Leyte

What is the most important part in the solutions of solving problems in dynamics of motion? Free-body diagram or FBD

A vector diagram showing all external forces acting on a body

Baybay City, Leyte

What is the acceleration of the ball if air friction is negligible?

w = mg

ΣFy = ΣFΣF a

mg = ma

FBD

a = g = 9.8 m/s2

from FBD from Second Law

+y

Baybay City, Leyte

θ

μk L

What is the acceleration of the ball if air friction is negligible?

ΣFx = ΣF

+mgsinθ -f = ma

a = g(sinθ-μkcosθ)

from FBD from Second Law

mg

+y

+x

+N-f

+mgsinθ

-mgcosθ

+mgsinθ -μkN= ma+mgsinθ –μkmgcosθ = ma

aΣF

Baybay City, Leyte

What about this?

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mA=

mB=

+x

+y

+mAg

-T

+y

+x

+mBg

+mBgcosø+T

-mBg.sinθ

a ΣFA

aΣFB

-μmBg.cosθ

What about this?

Shows complicated FBD!

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Student’s response on FBD.

Wrong!Wrong!

Wrong!

Wrong!

Wrong!Correct!

Misconceptions on Free-Body diagram

Problem given to students…..

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This paper presents a generalized mathematical formula that can be used in solving problems of dynamics on one-dimensional motion of two-object system which was derived from the principles of Newton’s Laws of Motion.

The formula helps the difficulty of students in solving the problems because the solutions process escapes FBD.

Baybay City, Leyte

BA

AAB

mm

m

cossincossinBmga

Generalized formula for dynamics on two-object system

θ β

μAmA

mB

μBmBmA

Atwood-machine

mB

mA

β

mA

mB

mB

mA

mB

mA

mB

mA

mBmAmBmA

mB

mA

mB

mA

Baybay City, Leyte

BA

AAB

mm

m

cossincossinBmga


θ β

μAmA

mB

μB

Assumptions:• maximum of one pulley and it’s a frictionless.• Air friction is negligible.• motion is due to gravity only.• each object moves on a straight line.• object B accelerates down the plane. If calculated a

is negative, then it moves up the plane.• Neglect the effects of rotation of pulley & masses.• Negligible weight of cord, and no elongations.

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Baybay City, Leyte

BA

AAB

mm

m

cossincossinBmga


θ β

μAmA

mB

μB

Assumptions:• maximum of one pulley and it’s a frictionless.• Air friction is negligible.• motion is due to gravity only.• each object moves on a straight line.• object B accelerates down the plane. If calculated a

is negative, then it moves up the plane.• Neglect the effects on rotation of pulley & masses.• Negligible weight of cord, and no elongations.

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Baybay City, Leyte

θ β

μA

mA

μB

+y

+x

mBg

N -T

mBg.sinβ

aΣFB

-μBmBg.cosβ

+y

+x

+mAg

-μAmAgcosθ

+T

-mAg.sinθ

aΣFB N

-mBg.cosβ-mAg.cosθ

Derivation of Formula

mB

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+y

+x

mBg

N -T

mBg.sinβ

aΣFB

-μBmBg.cosβ

-mBg.cosβ

+y

+x

+mAg

-μAmAgcosθ

+T

-mAg.sinθ

aΣFA N

-mAg.cosθ

FBD of mA FBD of mB

ΣFx =

+T–mAgsinθ-μAmAgcosθ=mAa (eq.1)

ΣFx =

-T+mBgsinβ-μBmBgcosβ=mBa (eq.2)

Adding equations 1& 2

+mBgsinβ-μBmBgcosβ –mAgsinθ-μAmAgcosθ =mBa +mAa g[mB(sinβ-μBcosβ –mA(sinθ+μAcosθ )]=a(mB +mA)

BA

AAB

mm

m

cossincossinmg

a B

mAa mBaƩFy =0 ƩFy =0

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Common problems on Two-Object system

mBmA

Atwood-machine

mB

mA

mA

mB

BA

AAB

mm

m

cossincossinBmga

mB

mA

mB

mA

OR

mBmAmBmA

OR

Next

mB

mAmB

mA

OR

Assump

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